On the Commutant of Multiplication Operators with Analytic Rational Symbol

Abstract

Let ${\cal B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain $G$ in the complex plane. Let $\varphi$ be an analytic multiplier of ${\cal B}$ we denote by $M_{\varphi}$ and $\{M_{\varphi}\}^{'} $ respectively, the operator of multiplication by $\varphi$ and the commutant of $M_{\varphi}$. In this article under certain conditions on $\varphi$ and $G$ we characterize the commutant of $M_{\varphi}$. In particular, when $\varphi$ is a rational function with poles off $\overline G$, under certain conditions on $\varphi$ we show that $\{M_{\varphi}\}^{'} =\{M_{z}\}^{'} $. We extend some results obtained in [4] and [6] about the commutant of the operator $M_{\varphi}$.